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Digital Image Processing

Image Enhancement in the Spatial Domain

(Chapter 4)

The principal objective of enhancement is to process an images so that the result is more suitable than the original image for a SPECIFICapplication

Category of image enhancement• Spatial domain• Frequency domain

Objective

Pixel neighborhood

g(x,y) = T [ f(x,y) ]

Point Processing,Gray-Level Transformation Function

s = T (r)

Contraststretching

Thresholding

g(x,y) = T [ f(x,y) ] 1

0

0

-1

0

0

0

0

0

T

f(x,y) g(x,y)

Mask Processing Filtering

Some Basic gray-Level transformation for image Enhancement

Basic Gray Level Transformation

Image Negative

s = L - 1 - r

Log Transformations

s = c log(1+r)

Pixel values dynamic range=[0 - 1.5×106]

Power-Law Transformations

s = cr

s = r2.5

r = [1 10 20 30 40 210 220 230 240 250 255]

s( = 2.5) = [0 0 0 1 2 157 176 197 219 243 255]

s( =.4) = [28 70 92 108 122 236 240 245 249 253 255]

Power-Law Transformations - Gamma Correction

Power-Law Transformations

c=1

= 0.4c=1

= 0.3

c=1

= 0.6

Power-Law Transformations

c=1

= 3

c=1

= 5

c=1

= 4

•Advantage–Arbitrarily complex

•Disadvantage–More user input

•Type of Transformations–Contrast stretching

–Gray-level slicing

–Bit-plane slicing

Piecewise-Linear Transformation Functions

Contrast Stretching

Objective: Increase the dynamic range of the gray levels

Causes for poor image-Poor illumination-Lack of dynamic rangein the imaging sensor-Wrong lens aperture

Example 1For image with intensity range [50 - 150]What should (r1,s1) and (r2,s2) be to increase the dynamic range of the image to [0 - 255]?

r

s

Contrast Stretching

Objective: Highlighting a specific range of gray levels in an image.

Gray-Level Slicing

255 138 30

65 12 201

180 111 85

255

138

30

65

12

201

183

111

85

1 1 1 1 1 1 1 1

0 1 0 1 0 0 0 1

1 1 1 1 1 0 0 0

1 0 0 0 0 0 1 0

0 0 1 1 0 0 0 0

1 0 0 1 0 0 1 1

1 1 1 0 1 1 0 1

1 1 1 1 0 1 1 0

1 0 1 0 1 0 1 0

MSBLSB

1 1 0

0 0 1

1 0 0

MSB plane

1 0 1

1 0 1

1 1 1

LSB plane

Bit-Plane Slicing

8-bit fractal image used for Bit-Plane Slicing

Bit-Plane Slicing

HistogramThe histogram of a digital image with gray levels in the range [0, L-1] is a discrete function

h(rk) = nk, where rk is the kth gray level and nk is the number of pixels in the image having gray level rk.

Normalized HistogramDividing each value of the histogram by the total number of pixels in the image, denoted by n.

p(rk) = nk /n.Normalized histogram provide useful image statistics.

Histogram Processing

function Normalized_Hist = Img_Hist(img)

[R,C]=size(img);

Hist=zeros(256,1);

for r = 1:R

for c=1:C

Hist(img(r,c)+1,1)=Hist(img(r,c)+1,1)+1;

end

end

Normalized_Hist = Hist/(R*C);

plot(Normalized_Hist)

Histogram Extraction using Matlab

Histogram Processing

Histogram equalization is used to enhance image contrast and gray-level detail by spreading the histogram of the original image.

s = T ( r ) 0 r 1, where r and s are normalized pixel intensities

Conditions for the transformation(a) T( r ) is single-valued and monotonically increasing in the

interval 0 r 1(b) 0 T ( r ) 1 for 0 r 1

1

Histogram Equalization

Objective of histogram equalizationTransform the histogram function of the original image pr(r) to a uniform histogram function.

ps(s) = 1 0 s 1

Equalizer

Transformationpr(r) ps(s)

r

r dwwprTs

0

)()(

k

j

jrkk rprTs0

)()(

Continuous case Discrete case

Histogram Equalization

0

1

2

3

:

:

128

129

130

:

:

253

254

255

0

0

:

0

:

:

0.004

0.15

0.05

:

:

0.005

0.006

0.004

pr(rk)k

T (rk)

0

0

:

0

0

0

0.004

0.154

0.204

:

:

0.989

0.994

1

sk

0

0

0

0

:

:

10

39

52

:

:

252

253

255

0

0.004

0.008

0.011

:

:

0.5

0.505

0.51

:

:

0.992

0.996

1

rk

/255 *255

Input image Output image

Histogram Equalization

255

Histogram Equalization

Is histogram equalization a good approach to enhance the image?

Histogram Matching

0.6

0.12

0.08

0.0

:

0.01

0.005

0

0.6

0.72

0.8

0.8

:

0.9

0.95

1

153

184

204

204

:

230

242

255

Histogram Equalization

pr(rk) sks

Histogram Matching method generates a processed image that has a specified histogram.

Input image

pr(r)

T (r)

r

r dttp

0

)( s

Desired

Output image

pz(z)

G (z)

z

z dttp

0

)( v

s v, since both will have

approximately uniform pdf

z = G-1[T( s )] s = G[T( r )]

Histogram Matching

Histogram Matching

1) Modify the histogram ofthe image to obtain pz(z).

2) Find transformation function G (z) using themodified histogram instep 1.

3) Find the inverse G-1 (z)

4) ApplyG-1 to the pixels of the histogram-equalized image.

G (z)

G-1 (z)

Histogram Matching

125

128

0.98

0.002

250

2550.7

0.3

178

255

Local Histogram

Global Histogram

125 128

Local Histogram Enhancement

Local Histogram Enhancement

Contrast manipulation using local statistics, such as the mean and variance, is useful for images where part of the image is acceptable,but other parts may contain hidden features of interest.

Histogram Statistics for Image Enhanc.

Let (x,y) be the coordinates of a pixel in an image, and let Sxy

denote a neighborhood (sub-image) of specified size, centered at(x,y).

.)(][

)(

),(

,2

,2

),(

,,

xy

xyxy

xy

xy

Sts

tsStsS

Sts

tstsS

rpmr

rprm

The local mean and variance are the decision factors to whether applylocal enhancement or not.

Histogram Statistics for Image Enhanc.

otherwise ),(

AND if ),(.),(

210

yxf

DkDkMkmyxfEyxg

GSGGS xyxy

MG: Global meanDG: Global standard deviationE, k0, k1, k2: Specified parameters

E = 4,

k0 = 0.4,

k1= 0.02,

k2 = 0.4

Size of local area = 3 3

Histogram Statistics for Image Enhanc.

125 135 145 135 ..

168 175 158 149 ..

210 231 215 129 ..

187 192 145 200 ..

174

Image Mean

164

187

(x,y) (x,y) E

Histogram Statistics for Image Enhanc.

Enhanced ImageOriginal Image

Histogram Statistics for Image Enhanc.

Arithmetic/logic operations involving images are performed on a pixel-by-pixel basis between two or more images.

Arithmetic OperationsAddition, Subtraction, Multiplication, and Division

Logic OperationsAND, OR, NOT

Enhancement Using Arithmetic/Logic Op.

AND

OR

Logic operations are performed on the binary representation of the pixel intensities

Enhancement Using AND and OR Logic Op.

Enhancement Using Arithmetic Op. _ SUB

Original4

Upper-order

Bit

ErrorHE of

Error

Problem:The pixel intensities in the difference image can range from -255 to 255.Solutions:1) Add 255 to every pixel and then divide by 2.2) Add the minimum value of the pixel intensity in the difference image

to every pixel and then divide by 255/Max. Max is the maximumpixel value in the modified difference image.

Enhancement Using Arithmetic Op. _ SUBMask Mode Radiography

2

),(

2

),(

1

1

),(),(

),(1

),(

),(),(),(

yxyxg

k

i

i

k

yxfyxgE

yxgk

yxg

yxyxfyxg

Original image Noise with zero mean

Enhancement Using Arithmetic Op. Averaging

Gaussian Noise

mean = 0

variance = 64

K = 8

K = 32

K = 16

K = 128

f(x,y) g(x,y)

+-

x

Enhancement Using Arithmetic Op. Averaging

Difference images between original image and images obtained from averaging.

K = 8

K = 16

K = 32

K = 128

Notice the mean and varianceof the difference imagesdecrease as Kincreases.

Enhancement Using Arithmetic Op. Averaging

Spatial filtering are filtering operations performed on the pixel intensities of an image and not on the frequency components of the image.

a

as

b

bt

tysxftswyxg ),(),(),(

a = (m - 1) / 2 b = (n - 1) / 2

Basics of Spatial Filtering - Linear

Response, R, of an m n mask at any point (x, y)

mn

i

iizwR1

Special consideration is given when the center of the filterapproach the boarder of the image.

Basics of Spatial Filtering

Nonlinear spatial filters operate on neighborhoods, and the mechanics of sliding a mask past an image are the same as was just outlined. In general however, the filtering operation is based conditionally on the values of the pixel in the neighborhood under consideration, and they do not explicitly use coefficients in the sum-of products manner described previously.

ExampleComputation for the median is a nonlinear operation.

Nonlinear of Spatial Filtering

Smoothing filters are used- Noise reduction- Smoothing of false contours- Reduction of irrelevant detail

Undesirable side effect of smoothing filters- Blur edges

Weighted

average

Box

filter

Weighted average filterreduces blurring in the

smoothing process.

Smoothing Spatial Filtering - LinearAveraging (low-pass) Filters

n = filter size

n = 3

n = 5 n = 9

n = 15

n = 35

Smoothing Spatial Filtering _ LinearAveraging (low-pass) Filters

filter size

n = 15

Thrsh = 25% of

highest intensity

Smoothing Spatial FilteringAveraging & Threshold

Order-statistics filters are nonlinear spatial filters whose response is based on ordering (ranking) the pixels contained in the image area encompassed by the filter, and then replacing the value of the center pixel with the value determined by the ranking result.

3 3 Median filter [10 125 125 135 141 141 144 230 240] = 141

3 3 Max filter [10 125 125 135 141 141 144 230 240] = 240

3 3 Min filter [10 125 125 135 141 141 144 230 240] = 10

Median filter eliminates isolated clusters of pixels that are light or dark with respect to their neighbors, and whose area is less than n2/2.

Smoothing Spatial FilteringOrder Statistic Filters

n = 3

Average

filter

n = 3

Median

filter

Order Statistic Filters

The principal objective of sharpening is to highlight fine detail in an image or to enhance detail that has been blurred.

9

1

9

1

9

1

9

1

z

zi

Image Blurred Image

The derivatives of a digital function are defined in terms of differences.

Sharpening Spatial Filters

Requirements for digital derivativeFirst derivative

1) Must be zero in flat segment2) Must be nonzero along ramps.3) Must be nonzero at the onset of a gray-level step or ramp

Second derivative1) Must be zero in flat segment2) Must be zero along ramps. 3) Must be nonzero at the onset and end of a gray-level step

or ramp

)(2)1()1(

)()1(

2

2

xfxfxfx

f

xfxfx

f

Sharpening Spatial Filters

Sharpening Spatial Filters

Comparing the response between first- and second-ordered derivatives:1) First-order derivative produce thicker edge2) Second-order derivative have a stronger response to fine detail, such as thin lines and isolated points.3) First-order derivatives generally have a stronger response to a gray-level step {2 4 15}4) Second-order derivatives produce a double response at step changes in gray level.

In general the second derivative is better than the first derivative for image enhancement. The principle use of first derivative is for edge extraction.

Sharpening Spatial Filters

First derivatives in image processing are implemented using the magnitude of the gradient.

yx

t

GGy

f

x

fmagf

y

f

x

f

5.022

)( f

f z1 z2

z6

z8

z4

z7

z3

z9

z5

Roberts operator

Gx = (z9-z5) and Gy = (z8 - z6)Sobel operator

Gx = (z7+2z8+z9) - (z1+2z2+z3) and

Gy = (z3+2z6 +z9) - (z1+2z4+z7)

Use of First Derivative for Edge Extraction Gradient

Sobel operators

Robert operator

Use of First Derivative for Edge Extraction Gradient

f(x,y) = [40, 140]

…. 0 0 0 0 0 0

…. 0 200 400 400 400 400 ……

…. 0 400 600 400 400 400 …...

…. 0 400 400 0 0 0

…. 0 400 400 0 0 0

…. 0 400 400 0 0 0

-1 -2

0

2

0

1

-1

1

0

-1 0

2

0

-2

-1

1

1

0

… 0 0 0 0 0

… 0 100 200 200 200

… 0 200 0 0 0

… 0 200 0 0 0

-11

-11

Use of First Derivative for Edge Extraction Gradient

Use of First Derivative for Edge Extraction Gradient

2

2

2

22

y

f

x

ff

)],(4)1,()1,(),1(),1([

),(2)1,()1,(

),(2),1(),1(

2

2

2

2

2

yyfyxfyxfyxfyxff

yxfyxfyxfy

f

yxfyxfyxfx

f

2nd Derivative _ Laplacian

2

2

2

22

y

f

x

ff

Isotropic filter response is independent of the direction of the discontinuities in the image to which the filter is applied.

Use of 2nd Derivative for EnhancementLaplacian

f(x,y) = [90, 100]0 1

1

1

1

0

0

0

-4

… 0 0 0 0 …

… 0 10 0 0 …

… 10 -40 10 0 …

… 0 10 0 0 …

… 0 0 0 0 …

Use of 2nd Derivative for EnhancementLaplacian

),(),(

),(),(),(

2

2

yxfyxf

yxfyxfyxg

If the center coefficientof the laplacian mask isnegative

1 1

1

1

1

1

1

1

-8

Use of 2nd Derivative for EnhancementLaplacian

g(x,y) = f(x,y) -

0 1

1

1

1

0

0

0

-4

),(2 yxf

=

)],(4)1,()1,(),1(),1([2 yxfyxfyxfyxfyxff

Use of 2nd Derivative for EnhancementLaplacian

High-boost filtering is used when the original image isblurred and dark.

),(),( 2 yxfyxAffhb A > 1

Un-sharp Masking and High-boost Filtering

(b)

Un-sharp Masking and High-boost Filtering

Combining Spatial Enhancement Methods

Combining Spatial Enhancement Methods